The strange case of x/x=y Is there a term for this?

[Bohandas]

I noticed that some of the normal rules of algebra don't technically apply x/x=y and related terms and I wondered which branch of math would deal with things like this and if there's a special word for the fact that they're not equivalent.

Specifically x/x=y is defined for every value of x except for 0 and for exactly one value of y (y=1). Conversely, the statement x=xy is defined for every value of x and for every value of y (for example if we are told that y=4327 then x must equal zero, or if we are told that x=-60 then y must be equal to 1). Furthermore, neither of these are precisely the same as x/y=x, which is mostly the same as x=xy except that it is undefined when y=0. Despite the fact that at the level of highschool algebra these ought to be equivalent (if we take x/x=y and multiply both sides by x we get x=xy, and if we then divide both sides by y we get x/y=x)

Is there a word for the fact that these aren't equivalent and/or a branch of mathematics that deals with things like this?

[tyckspoon]

That.. mostly sounds like you're asking if there's a particular branch of mathematical study concerning the fact division by zero is undefined? Because that appears to be the common factor in all of those - the equivalencies only work if one of the values is either zero or 1; in the case of zero it's a known exception to normal rules, and in the case of 1 it's functionally not even in the equation at all, because it has no effect on the results, so you may as well say 'x = x, how weird is that?'

[jayem]

You have analytic continuation, that basically deals with defining undefined values in a sensible fashion.

Basically it's to do with having a well defined multiplicative inverse (division). The difference between rings and fields is basically this (when it occurs for cases that aren't zero).
In mod 12, you have a worse problem. 3*0=0, 3*4=0, 3*8=0. So 0/3 could be 0, 4 or 8
You can't escape the additive identity (0) being like that. In something that has nice addition and multiplication

So I'd be longing at something like invertible functions.

[Radar]

You have analytic continuation, that basically deals with defining undefined values in a sensible fashion.

Basically it's to do with having a well defined multiplicative inverse (division). The difference between rings and fields is basically this (when it occurs for cases that aren't zero).
In mod 12, you have a worse problem. 3*0=0, 3*4=0, 3*8=0. So 0/3 could be 0, 4 or 8
You can't escape the additive identity (0) being like that. In something that has nice addition and multiplication

So I'd be longing at something like invertible functions.

Even before the inverse there is this property:

a*(b+c)=a*b+a*c

from which the problems of 0 (neutral element of addition) come from, since inevitably

a*b=a*(b+0)=a*b+a*0
0*a=0

[Diceomancer]

This can be handled with a rule from calculus called L'Hôpital's rule. It will also come out to y=1 at x=0.

https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule

[Caerulea]

Even before the inverse there is this property:

a*(b+c)=a*b+a*c

from which the problems of 0 (neutral element of addition) come from, since inevitably

a*b=a*(b+0)=a*b+a*0
0*a=0

I do not see the issue with this. So long as you don't attempt to invert the process (i.e. 'factor' 0 in to a * 0) then what is the problem?

—Caerulea

[jayem]

Even before the inverse there is this property:

a*(b+c)=a*b+a*c

from which the problems of 0 (neutral element of addition) come from, since inevitably

a*b=a*(b+0)=a*b+a*0
0=0*a

Yep it's the distributive property that ensures it (combined with the use of negation and addition (subtraction) to get rid of the a*b.
Though that just shows that 0*a=0 (for any 'a' and for anything interestingly 0 like), which isn't a problem until you expect a unique 'a' for which this is true or try to find non-zero a*b=0 (I.E get division involved anywhere). [I.E. to deal with X-post It's a two part problem, something with that feature, clearly has 'problems', and this shows that zero can't avoid that feature]


Regarding L'Hôpital's rule
I think at even higher levels of calculus the phrasing has to be a bit more cautious (although it probably depends on who you talk too). You say something more like y tends to 1 as x tends to 0. And then you say, "and anyway I'm not talking about y=x/x I'm talking about the function that is defined as y=x/x if x!=0 and y=1 if x=1"
Which is basically a complex way of not admitting what you are doing.

[Radar]

Yep it's the distributive property that ensures it (combined with the use of negation and addition (subtraction) to get rid of the a*b.
Though that just shows that 0*a=0 (for any 'a' and for anything interestingly 0 like), which isn't a problem until you expect a unique 'a' for which this is true or try to find non-zero a*b=0 (I.E get division involved anywhere). [I.E. to deal with X-post It's a two part problem, something with that feature, clearly has 'problems', and this shows that zero can't avoid that feature]

Technically you don't even need subtraction, since
a*b=a*b+a*0
is basically this:
for all x: x=x+a*0
which makes a*0=0 by definition, unless a*b does not cover some possible values.

[Knaight]

The closest term I can think of for what you're describing here is "removable discontinuity". That's more the specific point than the function that has it, but at that point you just switch to "functions with removable discontinuities" and you're good. The specific case of y=x/x would put that discontinuity at 0 and be in a straight horizontal line at 1 for all other points, but there's all sorts of otherwise smooth curves that can have these.

[wumpus]

Oddly enough, the function you are griping about is extremely similar to a delta function: one that is infinite at zero, and zero everywhere else (and integrates to 1 for any integral that crosses 0). It is extremely important to signal processing (most filters are defined by their response to this function) and pops up in a lot of similar math (especially anything similar to a Fourier transform).

[maruahm]

Mathematician here. The function f(x)=x/x is, as OP noted, properly defined only for ℝ∖{0}=(-∞,0)∪(0,∞). This isn't a strange case but just an example of a function's definition restricting its domain in order to be well-defined, meaning that it makes sense to talk about evaluating the outcome of the function.

jayem also accurately notes that there is an analytic continuation of the function, and it is... drumroll please... f*(x)=1. This "plugs the hole" at x=0 so that f is continuous and infinitely differentiable.

Oddly enough, the function you are griping about is extremely similar to a delta function: one that is infinite at zero, and zero everywhere else (and integrates to 1 for any integral that crosses 0). It is extremely important to signal processing (most filters are defined by their response to this function) and pops up in a lot of similar math (especially anything similar to a Fourier transform).

The delta function is an engineering and physics simplification of what in math is called a distribution. That's because, obviously, no function can take an infinite value in a way that's properly defined. You'll also find that the Riemann integral for the delta function doesn't work, since a full measure theoretic treatment is needed to rigorously talk about distributions like the delta function.